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On the Smoothness of Value Functions

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  • Strulovici, Bruno
  • Szydlowski, Martin

Abstract

We prove that under standard Lipschitz and growth conditions, the value function of all optimal control problems for one-dimensional diffusions is twice continuously differentiable, as long as the control space is compact and the volatility is uniformly bounded below, away from zero. Under similar conditions, the value function of any optimal stopping problem is continuously differentiable. For the first problem, we provide sufficient conditions for the existence of an optimal control, which is also shown to be Markov. These conditions are based on the theory of monotone comparative statics.

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File URL: http://mpra.ub.uni-muenchen.de/36326/
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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 36326.

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Date of creation: 07 Jan 2012
Date of revision: 31 Jan 2012
Handle: RePEc:pra:mprapa:36326

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Related research

Keywords: Super Contact; Smooth Pasting; HJB Equation; Optimal Control; Markov Control; Comparative Statics; Supermodularity; Single-Crossing; Interval Dominance Order;

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  1. Manuel Santos & Juan Pablo Rincon-Zapatero, 2007. "Differentiability of the Value Function without Interiority Assumptions," Working Papers 0704, University of Miami, Department of Economics.
  2. John Quah & Bruno Strulovici, 2007. "Comparative Statics, Informativeness, and the Interval Dominance Order," Economics Series Working Papers 2007-WO4, University of Oxford, Department of Economics.
  3. John K.-H. Quah & Bruno Strulovici, 2013. "Discounting, Values, and Decisions," Journal of Political Economy, University of Chicago Press, vol. 121(5), pages 896 - 939.
  4. Juan Pablo Rinc ́on–Zapatero & Manuel S. Santos, 2010. "Differentiability of the Value Function in Continuous–Time Economic Models," Working Papers 2010-28, University of Miami, Department of Economics.
  5. John K.‐H. Quah & Bruno Strulovici, 2012. "Aggregating the Single Crossing Property," Econometrica, Econometric Society, vol. 80(5), pages 2333-2348, 09.
  6. Cripps, Martin William & Keller, Godfrey & Rady, Sven, 2003. "Strategic Experimentation with Exponential Bandits," CEPR Discussion Papers 3814, C.E.P.R. Discussion Papers.
  7. Milgrom, P. & Shannon, C., 1991. "Monotone Comparative Statics," Papers 11, Stanford - Institute for Thoretical Economics.
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Cited by:
  1. Godfrey Keller & Sven Rady, 2013. "Breakdowns," Levine's Working Paper Archive 786969000000000635, David K. Levine.
  2. Martin Szydlowski, 2012. "Ambiguity in Dynamic Contracts," Discussion Papers 1543, Northwestern University, Center for Mathematical Studies in Economics and Management Science.

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